7s Mr. Ivory on the Attractions of an 
to the other, except in deducing the same conclusion with 
greater clearness and expressing it. with greater simplicity, 
and in a form better fitted to fulfil the views of the analyst. 
In these respects it can hardly be denied that the procedure 
delivered in the preceding pages has some advantages above 
that of the author of the Mecanique Celeste. The analysis here 
given is direct ; and it exhibits the several coefficients in sepa- 
rate and independent expressions derived immediately from 
the radius of the spheroid. On the other hand Laplace's 
investigation is indirect ; and the coefficients are found suc- 
cessively by decomposing the radius of the spheroid into a 
series of parts which follow a known law. If we now com- 
pare the two methods with respect to the grounds on which 
the investigations are founded we shall not find the same 
agreement between them. In this paper it is admitted as a 
necessary hypothesis, that the radius of the spheroid must be 
a rational and integral function of three coordinates of a point 
in the surface of a sphere : and, in consequence, the result of 
the analysis is limited to spheroids of that description. La- 
place, grounding his investigation on a property which, ac- 
cording to his demonstration, belongs to all spheroids that 
differ little from spheres, seems to prove that the radius of 
such a spheroid cannot be an arbitrary expression, and in this 
inference it is necessarily implied that the radius must be such 
a function as we have supposed it to be.* What in the one 
* Mec. Cel. Liv. $e, No. u. In No. n, by substitution in his fundamental theo- 
rem, Laplace obtains this formula 
.ay — 
U'(°) 3.eK 1 ^ 5.U (z) 
— — 4- e — 4- - — 
-J- Sit. : 
of this he remark*, a few lines below ; “ Cette expression de y n’est done point 
